The role of the mass matrix in the analysis of mechanical systems

Abstract

This paper proposes a simple, intuitive method of analyzing and understanding the mass matrix of a mechanical system. With very little calculation, it is possible to determine four important values that describe the inertia and inertial coupling of a mechanism: the locked effective inertia, the force coupling, the free effective inertia, and the acceleration coupling. These values can be determined for a system formulated in terms of joint coordinates, but when used with systems formulated in terms of workspace coordinates, this work provides a unified method of determining important, well-established concepts such as effective mass (particularly important in the field of human-robot interaction) and dynamic isotropy (an important consideration for robotic manipulators and haptic devices). Two examples are provided which highlight the use of this method to analyze robotic manipulators.

1 Introduction

Mass and inertia properties are of critical importance in analyzing the behavior of mechanical systems. This paper presents a method of revealing such information using simple analysis of the mass matrix. It is most easily illustrated here using robotic manipulators as examples, but can equally be applied to many other multibody systems.

The equations of motion for a multibody system expressed in a minimum set of independent coordinates are commonly written in the form

$$ \mathbf{M}(\mathbf{x})\ddot{\mathbf{x}} + \mathbf{c}(\mathbf{x}, \dot {\mathbf{x}}) = \mathbf{f}, $$

(1)

where M is the mass matrix, x is the vector of generalized coordinates, c is the vector of Coriolis and centrifugal terms, and f is the vector of generalized forces. The systems of equations can include both rotational or translational terms, and can be expressed at configuration, velocity, or acceleration level. This paper is primarily concerned with the mass matrix and how it relates to the movement of a mechanism in a particular coordinate direction, which is referred to here as a DOF (degree of freedom).

It is generally understood that the diagonal terms of the mass matrix are related to the inertias of the corresponding DOFs, and the off-diagonal terms express the inertial coupling between the DOFs. However, this explanation is insufficient, since it contains implicit assumptions are often overlooked. The inertias corresponding to the DOFs can vary greatly depending on the behavior of the remaining DOFs.

The two extreme cases are when the remaining DOFs are rigidly locked and prevented from moving (i.e., a motor and transmission that is not back-drivable or an actuator that is actively controlled to maintain a fixed position), or when the remaining DOFs are completely free to move (i.e., a perfectly free, frictionless joint). In order for the statement about the effective inertia of a joint to have any significance, the condition of the remaining DOFs must be specified.

The explanation that the off-diagonal terms express the inertial coupling between the DOFs is similarly incomplete. What exactly is the relation between the accelerations or forces corresponding to the DOFs? Does it matter what is happening at the other DOFs when considering this?

This paper proposes a simple method of interpreting the physical significance of the terms of the mass matrix, in a way that makes it possible to intuitively understand the behavior of the system and the coupling between the modes of movement, as well as provide corresponding mathematical relations.

The method produces four important concepts for analyzing a mechanism:

• the effective inertia of a mechanism in a particular direction when the remaining DOFs are locked (locked effective inertia),

• the force required to maintain the locked DOFs (force coupling)

• the effective inertia of a mechanism in a particular direction when there is no net force in the remaining DOFs (free effective inertia), and

• the acceleration in the direction of the unforced DOFs (acceleration coupling).

The significance of the values depends on the choice of coordinates. It is common in mechanical systems to choose relative coordinates that correspond to the joints of the mechanism. In this case, the inertia and coupling terms relate to the DOFs corresponding to the joints.

Other sets of coordinates are important for different applications. For example, a common practise in robotic systems is to formulate the equations in terms of the operational space coordinates, which are coordinates that are chosen to describe important aspects of the operation of the mechanism. While the operational space coordinates can be chosen to reflect any motion of the robot that is of interest, in most cases for manipulators, the three translations and three rotations of the end effectors are often chosen. In this case, the terms of the mass matrix take on a different significance. Rather than the inertia that is felt when actuating a particular joint, it describes the inertia of moving the end effector in a particular direction. This important measure is sometimes referred to as effective mass or reflected inertia.

In the operational space formulation, the coupling terms relate to the coupling between the operational space directions, which is an important consideration in the control of robots, as well as for the design of robots for certain applications, such as haptics. The coupling terms also make known the forces that are dynamically generated in the other coordinate directions when attempting a movement in a single coordinate direction.

One field where the mass matrix is investigated in detail is the area of vibrations. Multi-DOF oscillatory systems can be simplified into independent modes of vibrations using eigen-decomposition. This procedure is a mainstay of any modern textbook on linear mechanical systems [1] or vibrations [9]. However, this procedure is an analysis of the dynamic matrix, which involves both the mass matrix and stiffness matrix of a dynamic system.

There have been some attempts made to understand significance of the mass matrix, particularly for systems formulated in terms of the operational space of a mechanism. Asada [3] examined the mass matrix for a robotic manipulator in operational space coordinates and used ellipsoids to graphically represent the inertia of the system. He also investigated the concepts of dynamic isotropy and uncoupled systems, which is discussed in more detail in Sect. 3.2.

Khatib [8] also investigated the mass matrix for a robotic manipulator in operational space coordinates, but dealt primarily with the inverse of the mass matrix. He used three-dimensional shapes (belted ellipsoids) to represent the inertial properties. He also derived the concept of effective mass, which is described in more detail in Sect. 3.3.

Asada [3] and Khatib [8] also used graphical representations and eigenvalue decomposition in order to make sense of the seemingly complicated mass matrix for multibody systems. However, the method proposed here provides a direct explanation and interpretation of the terms of the mass matrix with no additional processing. This method also provides a unified system for investigating dynamic isotropy, coupling, and effective mass, as well as other concepts.

2 Formulation and concepts

For the remainder of the derivations, the vector of Coriolis and centrifugal terms will be combined with the generalized forces to form the vector f, and Eq. (1) simplifies to

$$ \mathbf{M}\ddot{\mathbf{x}} = \mathbf{f}. $$

(2)

The simplest case is when the mass matrix is diagonal, which means there is no inertial coupling between any of the generalized accelerations. In this case, it is obvious that each diagonal term of the mass matrix represents the effective inertia associated with the corresponding coordinate, i.e. m _1,1 is the inertia corresponding to coordinate x ₁.

If the mass matrix is not diagonal, there is coupling between the generalized accelerations, and a force in one coordinate direction will cause acceleration in the coupled directions also. This coupling is related to the off-diagonal terms of the mass matrix, but the magnitude is not obvious. Also, due to this coupling, the significance of the diagonal terms is no longer clear.

In order to extract meaning from the terms of the mass matrix, consider a system with three DOFs (assuming a symmetrical mass matrix, which is generally the case for systems with independent coordinates):

$$ \left[ \begin{array}{c@{\quad }c@{\quad }c} m_{1,1} & m_{1,2} &m_{1,3} \\ m_{1,2} & m_{2,2} &m_{2,3} \\ m_{1,3} & m_{2,3} &m_{3,3} \end{array} \right] \left[ { \begin{array}{c} \ddot{x}_1 \\ \ddot{x}_2 \\ \ddot{x}_3 \end{array} } \right]= \left[ { \begin{array}{c} f_1 \\ f_2 \\ f_3 \end{array} } \right]. $$

(3)

The equations are not easy to interpret unless special cases are considered. The first special case is when all DOFs are prevented from moving except one. If the motion in all of the directions except the first is locked (i.e., $\ddot{x}_{2}=0$ and $\ddot{x}_{3}=0$), the equations can easily be rearranged to

(4)

(5)

(6)

Equation (4) shows that the term m _1,1 directly corresponds to the effective inertia in the x ₁ direction when the remaining DOFs are locked.

The DOFs corresponding to x ₂ and x ₃ are prevented from moving, but there could be generalized forces associated with the DOFs. If the corresponding joints are physically locked, the generalized forces f ₂ and f ₃ could be considered reaction forces. Solving Eq. (4) for $\ddot{x}_{1}$ and substituting into Eqs. (5) and (6) results in

(7)

(8)

This means that if force f ₁ is applied in the direction of coordinate x ₁, these expressions determine the generalized forces that will be generated at the locked coordinates due to inertial coupling.

It is clear that for a simple system of equations that are linear in terms of accelerations as Eq. (2), the results of this three-DOF example can be generalized to any n-DOF system, where i,j={1,…,n}. This leads to the following concepts.

Definition 1

For a system of equations, m _i,i represents the effective inertia corresponding to generalized coordinate x _i when all other coordinate directions are locked (locked effective inertia):

$$ f_i = m_{i,i} \ddot{x}_i . $$

(9)

Definition 2

The relation between f _i , a force applied in the coordinate direction x _i , and the resulting force, f _j , generated in the locked coordinate direction, x _j (force coupling) is

$$ f_j = \frac{m_{i,j}}{m_{i,i}} f_i . $$

(10)

Additional useful concepts can be developed by rearranging Eq. (2) to form

$$ \ddot{\mathbf{x}} = \mathbf{W}\mathbf{f}, $$

(11)

where W is the inverted mass matrix, M ⁻¹.

Using the same three-DOF system, but this time specifying that the DOFs corresponding to x ₂ and x ₃ are “unforced” (f ₂=0 and f ₃=0), the equations can be expanded to

(12)

(13)

(14)

It can be seen from Eq. (12) that the effective mass in the x ₁ direction is $\frac{1}{w_{1,1}}$.

Solving Eq. (12) for f ₁ and substituting into Eqs. (13) and (14) results in

(15)

(16)

This means that if force f ₁ is applied in the direction of generalized coordinate x ₁, these expressions determine the accelerations that the remaining, unforced coordinates will undergo due to inertial coupling.

Again, due to the linearity of the equations in terms of accelerations, this can be generalized to any n-DOF system, where i,j={1,…,n}.

Definition 3

For a system of equations, $\frac{1}{w_{i,i}}$ represents the effective inertia corresponding to generalized coordinate x _i when there is no net force in all other coordinates directions (free effective inertia):

$$ f_i = \frac{1}{w_{i,i}}\ddot{x}_i . $$

(17)

Definition 4

The relation between acceleration $\ddot{x}_{i}$, and the accelerations in the free, unforced coordinate directions, x _j , when a force f _i is applied in the coordinate direction x _i (acceleration coupling) is

$$ \ddot{x}_j = \frac{w_{i,j}}{w_{i,i}} \ddot{x}_i . $$

(18)

The terms “free” and “unforced” here refer to a DOF with no associated generalized force, or where the net sum of the forces is zero.

It is worth noting that although the preceding concepts are easiest to understand and apply to mechanisms at rest (no velocity, and hence no Coriolis and centrifugal forces), it is not only valid in those cases. The expressions relating to the generalized forces at the DOFs, f _i , include the external and applied forces and Coriolis and centrifugal terms, and the formulation is equally valid for mechanism in motion or with external forces such as gravity. In cases where the DOFs are considered “free”, it is required that there be no net force at the DOF, which includes cases where there are no forces, or cases when an actuator is actively controlled to counteract the external forces and Coriolis and centrifugal terms.

In cases referring to “locked” DOFs, it is only strictly necessary that the system undergo no accelerations in the DOF directions, which includes the case where the velocities in the DOF directions are constant.

3 Analysis and applications

There are many immediately useful quantities that can be determined from the mass matrix in a joint space formulation, but the method provides additional information when the equations of motion are transformed into different sets of coordinates, such as operational space coordinates. The interpretation of the mass matrix outlined above still holds, but provides a different significance to the values. All of the combined possibilities are outlined in Table 1. This section examines several of these aspects in more detail and suggests some possible applications.

Table 1 Indicators interpreted from the mass matrix

This formulation is valid for any system represented in a minimum set of independent coordinates. While this includes many vehicles, biological systems, and other mechanical systems, the concepts are illustrated here with serial robotic manipulators.

3.1 Locked effective inertia and force coupling of joints

The concepts of locked effective inertia and force coupling are interesting because they are read directly from the mass matrix with no additional calculation. This makes them particularly useful to quickly understand the dynamic behavior and interaction of the various DOFs of a mechanism.

For a serial manipulator formulated in terms of relative joint coordinates, the locked effective inertia indicates the inertia that must be overcome in order to actuate a joint, when the remaining joints are locked. The force coupling terms indicate the amount of force or torque that would be felt by the other joints, which could also be interpreted as the amount of force or torque that must be generated in order to maintain the joints locked.

Another useful example would be the wheels of a rover. The locked effective inertia of a wheel joint is the inertia felt by the wheel actuator. The force coupling terms indicate the forces and torques that would result at the other joints. This would indicate the torque that would be caused in the suspension when the wheel is actuated, or the pitch and roll torque felt by the entire body of the rover.

3.2 Locked effective inertia and force coupling of operational space directions

For a serial manipulator formulated in terms of operational space coordinates, the locked effective inertia is an indication of the force required in a particular operational space direction, when the other directions are constrained. For example, if the operational space coordinates are the x and y directions of a flat workspace, the locked effective inertia in the x direction indicates the inertia that would resist motion in the x direction if the end effector is prevented from moving in the y direction.

In the same example, the force coupling terms indicate the reaction force that would result (or must be generated by the actuators) in the y direction in order to produce a straight motion in the x direction.

This also relates to the concepts of inertial decoupling (no inertial coupling between the operational space coordinates) and dynamic isotropy (the effective mass is equal in all operational space directions). These are very important criteria for many devices, such as haptic systems where it is important that when a mechanism is moved in one direction, it does not generate extraneous forces in other directions.

The concept of determining an effective end effector mass has been investigated in the past [3]. Asada [3] defined the generalized inertia matrix (referred to in this paper as the operational space mass matrix) for a serial manipulator, M _t , as

$$ \mathbf{M}_t = \mathbf{R}^\mathrm{T}\mathbf{M}\mathbf{R}, $$

(19)

where M is the mass matrix of the manipulator in joint coordinates and R is the inverse of the Jacobian matrix associated with the transformation from joint coordinates to operational space coordinates.

Asada [3] used the fact that a mechanism is isotropic when the eigenvalues of M _t are equal. For the two-dimensional manipulator analyzed in the paper, this occurs when m _t1=m _t2 and m _t3=0 where

$$ \mathbf{M}_t = \left[ \begin{array}{c@{\quad }c} m_{t1} & m_{t3} \\ m_{t3} & m_{t2} \end{array} \right]. $$

(20)

The same result follows from the method proposed in this paper: For a decoupled, isotropic system, the coupling terms must be zero and the effective inertia terms must be equal.

It is important note that the locked effective inertia in a particular direction is dependent on the entire formulation, since the representation use for the remaining DOFs determines which directions of movement are fixed.

3.3 Free effective inertia of operational space directions

The free effective inertia of an operational space direction is equivalent to the reflected inertia of an end effector used in robot design and analysis [6]. It is the same concept as the effective mass proposed by Khatib [8].

Khatib [8] also defines a matrix (M _t ) equivalent to the operational space mass matrix:

$$ \mathbf{M}_t^{-1}(\mathbf{x}) = \mathbf{R}(\mathbf{x})\mathbf {M}^{-1}(\mathbf{x})\mathbf{R}^\mathrm{T}(\mathbf{x}), $$

(21)

where M is the mass matrix of the manipulator in joint coordinates and R is the Jacobian matrix associated with the transformation from joint coordinates to translational operational space coordinates. All are dependent on the system configuration, represented by x.

While Asada [3] primarily analyzed the mass matrix, Khatib [8] focused on the properties of the inverse of the mass matrix and defined the effective mass. Khatib [8] showed that the effective mass of an end effector (m _u ) can be calculated in any direction from

$$ \frac{1}{m_\mathbf{u}(\mathbf{M}_t)} = \mathbf{u}^\mathrm{T}\mathbf {M}_t^{-1}(\mathbf{x})\mathbf{u}, $$

(22)

where u is a unit vector in the desired direction, represented in terms of operational space coordinates. This allows the free effective inertia to be calculated for any unit vector, u, that can be described in terms of the operational space directions. A simplified case would be when the vector u directly corresponds to the direction associated with an operational space coordinate. In such cases, it is clear that the free effective inertias are simply the diagonals of the mass matrix, as was determined in Sect. 2.

The free effective inertia is particularly important in impact situations. One example is in the field of human-friendly robot design. In some situations, (where the human is constrained), the inertia at the tip of the robot (reflected inertia) is one of the most critical indicators of injury potential [5, 6]. While the full measure of the inertia of the robot is dependent on many factors, such as actuator design and control [10], the free effective inertia provides a very simple way of determining the contribution of the mechanism geometry, properties, and configuration to the inertia of a robot.

The free effective inertia also relates to the concept of inertial impedance, which is important for many systems, including human limbs [2]. It is also an indication of the unsprung mass for a vehicle or rover, which includes the behavior of the suspension system in addition to the wheel.

3.4 Acceleration coupling between operational space directions

In operational space coordinates, when a force is applied in one direction, the acceleration coupling terms indicate what accelerations will be generated in the remaining directions. For a serial manipulator, if a force is applied to the end effector, the movement is not generally in the direction of the applied force.

This also relates to the concept of decoupled systems discussed earlier. While the degree of coupling can be investigated through the force coupling terms and the mass matrix, it can also be determined from the acceleration coupling terms and the inverse mass matrix. If the mass matrix is diagonal, so is the inverse.

4 Examples

While the mathematics involved is simple, this method of analysis provides an easy way of understanding the behavior of any system expressed in minimum coordinates, simple or complex. In this section, several applications of the method will be investigated, including some simple examples, as well as some extensions to more complicated situations.

4.1 Simple serial manipulator

A two-DOF serial manipulator is an example where the equations are simple enough to be developed symbolically, yet it still illustrates the important concepts of the method. The system is shown in Fig. 1.

Fig. 1

Two-DOF serial manipulator

The system contains two rigid links connected with revolute joints. The two angles, θ ₁ and θ ₂ are the relative angles of the joints. The centres of mass are at the midpoints of the links. The parameters used are described in Table 2.

Table 2 Physical properties of two-DOF serial manipulator

The equations of motion can be derived and put into the form corresponding to Eq. (2) with

(23)

(24)

(25)

where T ₁ and T ₂ are the torques at the two joints and c ₁ and c ₂ are the Coriolis and centrifugal terms. For this example, it is assumed that the system starts from rest and c ₁=c ₂=0 in that instant.

For the configuration shown in Fig. 1 ($\theta _{1}=\frac{\pi}{3}$, $\theta_{2}=-\frac{\pi}{2}$) the mass matrix can be evaluated using the values in Table 2 to be

$$ \mathbf{M} = \left[ \begin{array}{c@{\quad }c} 1.92 & 0.32 \\ 0.32 & 0.32 \end{array} \right]\quad \mbox{kg}\,\mbox{m}^2. $$

(26)

It is then simple to interpret the terms of the mass matrix. According to Definition 1 (Eq. (9)), if θ ₂ is locked, the rotation of θ ₁ has an effective inertia of 1.92 kg m². Conversely, if θ ₁ is locked, the rotation of θ ₂ has an effective inertia of 0.32 kg m².

Definition 2 (Eq. (10)) can be used to calculate that if θ ₂ is locked and a torque T ₁ is applied at θ ₁, the torque required to maintain θ ₂ locked will be $\frac{0.32}{1.92}\,T_{1}$ or 0.167 T ₁. Conversely, if θ ₁ is locked and a torque T ₂ is applied at θ ₂, the torque required to maintain θ ₁ locked will be $\frac{0.32}{0.32}\,T_{2}$ or T ₂.

The inverse of the mass matrix can also be calculated.

$$ \mathbf{W} = \left[ { \begin{array}{c@{\quad }c} 0.625 & -0.625 \\ -0.625 & 3.75 \end{array} } \right]\quad \frac{1}{\mbox{kg}\,\mbox{m}^2}. $$

(27)

Definition 3 (Eq. (17)) states that if there is no net torque at θ ₂, the rotation of θ ₁ has a free effective inertia of 1.60 kg m². Conversely, if there is no net torque applied at θ ₁, the rotation of θ ₂ has an effective inertia of 0.267 kg m².

Definition 4 (Eq. (18)) can be used to calculate that if there is no net torque at θ ₂ and a torque T ₁ is applied at θ ₁, the acceleration of θ ₂ will be $\frac{-0.625}{0.625}\,\ddot{\theta}_{1}$ or $-\ddot{\theta}_{1}$. Conversely, if there is no net torque at θ ₁ and a torque T ₂ is applied at θ ₂, the acceleration of θ ₁ will be $\frac {-0.625}{3.75}\,\ddot{\theta}_{2}$ or $-0.167 \,\ddot{\theta_{2}}$.

The same method of simplifying and interpreting holds for systems formulated in terms of different coordinates, and in some cases the results can be very useful.

In this example, the equations for the two-DOF manipulator of Fig. 1 are transformed to operational space coordinates, as in Fig. 2. The transformation is performed using the procedure outlined in [7]. This results in a new, transformed mass matrix which can be written as

$$ \mathbf{M}_t = \mathbf{R}^{-\mathrm{T}}\mathbf{M}\mathbf{R}^{-1}, $$

(28)

where R is a Jacobian matrix that transforms from the joint coordinates to the operational space coordinates:

$$ \mathbf{R} = \left[ \begin{array}{c@{\quad }c} -l_2(\cos\theta_2 \sin\theta_1 + \sin\theta_2 \cos\theta_1)-l_1\sin\theta_1 & -l_2(\cos\theta_2 \sin\theta_1+\sin\theta_2 \cos\theta_1) \\[3pt] l_2(\cos\theta_2 \cos\theta_1-\sin\theta_2 \sin\theta_1)+l_1\cos\theta_1 & l_2(\cos\theta_2 \cos\theta_1-\sin\theta_2 \sin\theta_1) \end{array} \right]. $$

(29)

Fig. 2

Two-DOF serial manipulator operational space coordinates

For this configuration ($\theta_{1}=\frac{\pi}{3}$, $\theta_{2}=-\frac{\pi }{2}$) M _t can be calculated to be

$$ \mathbf{M}_t = \left[ \begin{array}{c@{\quad }c} 8.00 & -3.46 \\ -3.46 & 4.00 \end{array} \right]\quad \mbox{kg}. $$

(30)

As with the joint space mass matrix, there are several values that are directly useful. Definition 1 shows that the effective inertia in order to move directly in the x direction is 8 kg. This value provides a direct indication of the forces necessary to generate a given acceleration in the x direction, and could be used to estimate the necessary tuning values for the corresponding controller.

Definition 2 states that if a force, f ₁, is applied in the direction of positive x, a force of −0.433 f ₁ must also be generated in the y direction in order to maintain a purely horizontal motion for this instant. This directly provides the feed-forward value that would be used by the controller to maintain horizontal motion. As this is an operational space formulation, the acceleration and force values can be directly mapped to the joint actuator accelerations and torques using the Jacobian matrix R.

The inverse mass matrix of the transformed system can be evaluated for this configuration:

$$ \mathbf{W}_t = \left[ \begin{array}{c@{\quad }c} 0.200 & 0.173 \\ 0.173 & 0.400 \end{array} \right]\quad \frac{1}{\mbox{kg}}. $$

(31)

Based on Definition 3 it can be determined that the free effective inertia in the x direction is 5 kg. This important value is an indication of the forces that would be involved in a collision of the manipulator in the x direction.

It is also possible to evaluate the isotropy of a manipulator from the M _t matrix. Asada [3] determined that for an isotropic system, the effective mass terms (the diagonal terms of M _t ) should be equal and the coupling terms (the off-diagonal terms of M _t ) should be 0. He used this definition to calculate the dimensions and mass parameters that could be used to make a two-DOF manipulator that is dynamically isotropic in certain configurations. The parameters used are listed in Table 3.

Table 3 Physical properties of Asada’s two-DOF serial manipulator

Substituting these parameters and the configuration θ ₂=−90^∘, M _t can be evaluated to be

$$ \mathbf{M}_t = \left[ \begin{array}{c@{\quad }c} 39.7 & 0 \\ 0 & 39.7 \end{array} \right]\quad \mbox{kg}. $$

(32)

This clearly shows that the specified manipulator is isotropic in this configuration, as Asada [3] claimed.

In some cases, the mass matrices in this example were numerically evaluated, as symbolic expressions are often complicated to include. However, the method can be used to produce parametric expressions for the mass matrix elements. This means the expressions can be used as objective functions in order to optimize specific characteristics of the manipulator. For example, it might be desired to minimize the locked effective mass of a certain direction to increase manipulator acceleration, or reduce the non-diagonal terms of the mass matrix to reduce non-isotropic behavior, which is important for haptic devices.

4.2 Three-dimensional serial manipulator

The DLR¹ LWRIII (Light Weight Robot III) is a seven-DOF serial manipulator designed to provide large payload capacity with low overall mass, which makes it appropriate for applications with human interaction [4].

The robot investigated here is based on the LWRIII with the same topology and dimensions. However, since the mass properties are not publicly known, they were simply estimated based on the overall mass. The dimensions and mass properties are listed in Table 4. The robot and the seven rotational DOFs are as shown in Fig. 3.

Fig. 3

Seven-DOF serial manipulator based on the DLR LWRIII

Table 4 Physical properties of seven-DOF serial manipulator based on LWRIII

Haddadin et al. [6] performed several tests on four robotic manipulators, including the LWRIII to evaluate the dangers posed to nearby humans. They found that one of the most important factors determining the injury potential (particularly when the human is constrained) is the reflected mass in the direction that the robot strikes the human. This reflected mass is equivalent to the free effective mass, so the mass matrix analysis proposed here will naturally provide these values.

A full symbolic kinematic and dynamic model of the manipulator was generated using the Multibody Toolbox, a symbolic toolbox developed at the Canadian Space Agency using Maplesoft’s Maple software. The symbolic terms of the mass matrix are too lengthy to reproduce here, but the mass matrix was evaluated at various robot configurations to learn about the effective inertias and coupling of the various joints, as outlined in the previous sections.

As the mass is particularly of interest in a specific direction of the end effector, the system was transformed into the operational space. The transformation was performed at the velocity level, representing the system in terms of linear and rotational velocities corresponding to the three Cartesian directions of the end effector. If the full seven DOF system of equations were transformed into the six operational space velocities, the resulting Jacobian transformation matrix (R) would be nonsquare, and the resulting system of equations with six generalized velocities could not completely describe the state of the seven-DOF system. In order to avoid this situation, one of the joint velocities, corresponding to θ ₆, was considered a parameter that was assigned a constant value for various configurations. This reduces the system to six DOFs with a square Jacobian matrix.

The Jacobian matrix was formed from the symbolic kinematic model of the manipulator. The matrix was then numerically evaluated at each appropriate configuration, since symbolic matrix inversion would be too computationally costly. Equation (32) was then used to calculate the transformed mass matrix, M _t . This was then numerically inverted to form W _t . The free effective inertia was then simply $\frac{1}{w_{i,i}}$, where i is the index of the coordinate of interest, as stated in Definition 3.

The first analysis is performed for the “punching” motion performed in [6]. This motion can be seen in Fig. 4, where the end effector is moved horizontally in the direction of x by actuating joints θ ₂, θ ₄, and θ ₇.

Fig. 4

Seven-DOF manipulator in “punch” configuration

The value of the effective mass in the x direction changes significantly depending on the position of the end effector over the range of possible motion, as seen in Fig. 5.

Fig. 5

Free effective mass of end effector of seven-DOF manipulator in “punch” configuration

The manipulator investigated here cannot be quantitatively compared to the DRL LWRIII manipulator since the exact mass properties are unknown, but the relation between end effector position and effective mass in the x direction can be seen to be qualitatively very similar to that of Haddadin et al. [6, Fig. 3.d.], The effective mass changes little for the majority of the range, but near the full extension, it increases asymptotically. This is to be expected, since the manipulator approaches a singularity at full extension.

As the transformed mass matrix was already formed, the free effective masses in the other coordinate directions could be trivially calculated as well, and are also plotted in Fig. 5.

The second analysis is performed for the “swinging” motion considered in [6]. This motion corresponds to the configuration shown in Fig. 6, where the end effector is moved horizontally in the direction of y by actuating joint θ ₁. Note that the angle θ ₄ was set to 2° for this configuration, to avoid the singularity when the arm is fully extended.

Fig. 6

Seven-DOF manipulator in “swing” configuration

The free effective mass in the y direction can be calculated as before from the inverted operational space mass matrix, W _t . In the configuration shown in Fig. 6, it was found to be 2.8 kg. This provides an important indication of potential injury the robot could cause with this manoeuvre.

Since the mass matrix was already formed, it was also a simple matter to investigate the effective mass for a variation of the swing technique. The previous free effective mass was calculated for when the rotational axis of joint θ ₄ was parallel to θ ₁. If joints θ ₃ and θ ₅ are rotated through 90°, the rotational axis of θ ₄ is perpendicular to θ ₁. The free effective mass over the range of this rotation is plotted in Fig. 7.

Fig. 7

Free effective mass of end effector of seven-DOF manipulator in “swing” configuration

It can be seen that the free effective mass of the end effector is significantly higher when the axis of θ ₄ is perpendicular to θ ₁. This illustrates that the configuration of the entire manipulator must be considered when executing a manoeuvre.

The parametric expressions for the mass matrix terms are too complicated to reproduce here, but they could be used in optimization techniques. For this manipulator, reducing the free effective mass in various directions would be an important goal.

5 Conclusions

The method presented in this paper to analyze a mass matrix provides a simple, intuitive way of understanding the terms of the matrix. With no additional calculation, it is possible to see the effective masses corresponding to the DOFs of a system, and the coupling between the DOFs. While the concepts are simple, a clear explanation of the significance of these terms and corresponding mathematical relations are lacking in the multibody dynamics literature.

By itself, this is already useful to help a designer to come to an intuitive understanding of how a complicated multibody system works, or for an instructor to show a student how to interpret the terms of a system of equations. By performing a simple inversion of the mass matrix, another important set of values is determined.

All together, there are four main parameters that can be determined for a system of equations: the locked effective inertia, the force coupling, the free effective inertia, and the acceleration coupling. While these parameters already provide important information to interpret the behavior of a system in joint coordinates, the parameters reveal more information when the equations are formulated in terms of different coordinates.

In operational space coordinates, the parameters are directly related to the behavior of the system in the operational space directions, which is particularly useful in the area of robotic manipulators. The concepts of dynamic isotropy, inertial coupling, and effective mass are well developed and have been proven to be important analysis tools. The method proposed in this paper provides a unified method of determining and treating those values, and when the coordinate directions are correctly chosen, it can be simpler and more direct.

The method can also be used in Vortex or other general simulation software, provided that the mass matrix of the system can be exported from the software. However, as most general packages use global coordinates for each body (not a minimum set), the mass matrix would have to be projected onto the space of admissible motion that can be parameterized with minimum coordinates, which requires knowledge of the topology.

The analysis and examples in this paper were performed for robotic manipulators in joint space and operational space formulations since these are natural representations where the mass properties are critically important. However, the method can also be applied to any other system or formulation expressed in a minimum set of coordinates, which would provide additional inertia and coupling information.